Decidable objects and molecular toposes

We study several sufficient conditions for the molecularity/local-connectedness of geometric morphisms. In particuar, we show that if $\mathcal{S}$ is a Boolean topos then, for every hyperconnected essential geometric morphism $p:\mathcal{E}\to \mathcal{S}$ such that the leftmost adjoint $p_{!}$ preserves finite products, $p$ is molecular and $p^{\ast }:\mathcal{S}\to \mathcal{E}$ coincides with the full subcategory of decidable objects in $\mathcal{E}$. We also characterize the reflections between categories with finite limits that induce molecular maps between the respective presheaf toposes. As a corollary we establish the molecularity of certain geometric morphisms between Gaeta toposes.